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CHAPTER XVII.

EXCHANGE. THE CHAIN RULE.

1. THE value of a fixed sum of the money of one country expressed in that of another, when this value is calculated by a comparison of the weight and fineness of the coins of the two countries, is called the Par of Exchange. When for this fixed sum the correct equivalent in money of the other country, calculated on this supposition, can be obtained, the exchange is said to be at par. The exchange, however, between any two countries fluctuates from various causes.

The Course of Exchange is the variable sum of the money of one country which happens at a particular time to be equivalent to a fixed sum of the money of another country. Thus, at one time, for £1 25*15 francs, at another 25 20 francs, may be obtained, according to the course of exchange between England and France.

2. If A owes a correspondent in Berlin, for instance, £500, he might pay his debt by transmitting the value in coin or bullion. But this would be both cumbersome and expensive. If, then, he can find a person, B, who has money owing to him in Berlin, B can draw a bill upon his debtor in Berlin, and sell it to A, who then transmits it to his own correspondent in Berlin. Such bills of exchange are the means by which money transactions between different countries are conducted. The price of such bills will fluctuate according to the demand there may be for them at the time. The excess of their price over the sum they represent can clearly never exceed the amount of the cost of carriage and money value

J

of the risk incurred in forwarding the same sum in specie or bullion.

3. The Arbitration of Exchange is the process of fixing the rate or course of exchange between two places, by means of a comparison of the exchange between them and one or more intervening places. Thus a debt in Paris may be paid by means of a bill on Berlin, which is to be again replaced by one on Hamburg, and that, finally, by one from thence upon Paris. The arbitration is said to be simple when there is only one intermediate place, compound when there are more.

The subject of exchange, however, is too complicated for us to go into more than very superficially. For information we refer our readers to Kelly's "Universal Cambist."

A large number of questions in exchange can be worked out by the aid of the principles already laid down. Before proceeding to treat of them, we shall explain a method called the Chain Rule.

THE CHAIN RULE.

4. If the equivalent of any amount of one quantity is given in terms of another, that in terms of a third, and so on, it is required to find the equivalent of a certain amount of the first quantity in terms of the last.

EXAMPLE 1. -40 lbs. Troy of standard gold are coined into 1869 sovereigns, and standard gold contains II parts in 12 fine gold. Calculate the value of the money which can be coined out of 1 oz. of fine gold.

This contains

X 12 = weight of one sovereign in ounces.

X X 12 ounces of fine gold, which (neglect

ing the value of the alloy) are worth £1.

5. A convenient way of arranging the operation in questions of this kind is called the Chain Rule. It is especially useful in all questions connected with Exchange, and is the method generally used by merchants.

Write down the quantity of which the equivalent is required at the head of a column, as in the working given below. This quantity is generally called the Term of Demand.

Draw a sloping line from it to the left, at the extremity of which write down in a second column a quantity of the same kind as the first. Opposite to

this, place in the first column its equivalent in value of another quantity, and so on.

Thus, in the example already given, 1 oz. of fine gold is the term of demand. 11 oz. fine gold are equivalent to 12 oz. standard gold, 12 oz. standard gold are equivalent to 1 pound Troy standard gold, and 40 pounds Troy standard gold are equivalent to 1869 sovereigns; the last term in the right hand column being arranged to be sovereigns, because the answer is required in sovereigns.

Multiply together all the numbers in the longer column, and divide by the product of the numbers in the other column. This will give the equivalent of the term of demand in terms of the quantity standing lowest in the longer column (in the above example, sovereigns).

N.B. The sloping lines connect quantities of the same kind; the horizontal lines, quantities of equiva lent value.

6. The reason of the truth of this rule may be gathered from observing that in reality we multiply the term of demand by a succession of fractions which respectively express the value of one unit of a quantity in terms of the succeeding quantity. Thus we multiply the term of demand, which is 1 oz. fine gold, by 1, the number of ounces of standard gold equivalent to I oz. of fine gold. This gives us the amount of standard gold equivalent to 1 oz. of fine gold. We next multiply by, which expresses the same quantity in pounds Troy, and next by 1869, which is the equivalent of 1 pound Troy of standard gold in sovereigns.

For a more detailed explanation of the rule we refer our readers to Peacock's " Algebra," vol. i., arts. 346-353.

We give some additional examples worked out. The reader should carefully in each case examine the reason for the process in the way indicated above.

EXAMPLE 2.-If 7 lbs. of rice be worth 2 lbs. of currants, 3lbs. of currants 1 lb. of hops, 5 lbs. of hops 2 lbs. of tobacco at 48. per lb., what is the value of I lb. of rice ?

EXAMPLE 3.-A certain book is to contain 5 sheets of paper, and 4000 copies are to be printed. Supposing a ream of paper to weigh 24 lbs., what would be the saving upon the expense of publishing the work, owing to a reduction of the paper duty id. per pound?